Question

Which of the following expressions are polynomials
select all correct answers
A. 1+1/X
b.0
c.-XY-Z
d.y^x

Answer 1

c. and d.
a polynomial is a variable function which has positive values. n hey, dont forget the medal (y)

Answer 2

A polynomial has the general form
\[P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_2x^2+a_1x+a_0\]
where \(a_i\in\mathbb{C}\) for \(i=0,1,2,...,n\). Notice that \(n\) is a non-negative integer.

(a) can't be a polynomial because
\[1+\frac{1}{x}=1+x^{-1}\]
which has a negative exponent on one of the terms.

(b) is a polynomial - it consists of only a constant term, \(a_0\).

(c) can't be a polynomial because there is more than one variable. However, if \(y\) and \(z\) are kept fixed, you have linear expression:
\[-xy-z=a_1x^1+a_0\]
so it could be considered a polynomial. Similarly, if \(y\) is the variable and \(x\) and \(z\) are constant, you have another linear polynomial; the same is true if \(x\) and \(y\) are constant and \(z\) is the variable.

(d) is not a polynomial because there are multiple variables. However, if you fix \(x\) as a non-negative integer, you have a polynomial in terms of \(y\), like \(y^2\) or \(y^{10}\). Fixing \(y\) doesn't yield a polynomial, though.